In order to be able to transmit a digital signal, i.e. a stream of bits, via the medium of radio it is necessary to represent the bit stream as an analogue (continuous-time) signal. This involves the process of modulation. A technique commonly used for this purpose is passband pulse amplitude modulation (PAM) in which, in general, two sinusoidal carriers of the same frequency, but with a ninety degree phase difference, are modulated by the real and imaginary parts of a complex-valued baseband signal. The transmitter includes a coder which converts the incoming bits into so-called symbols, i.e. groups of bits which collectively constitute an alphabet.
At the receiver a demodulator is used to extract the discrete-time information from the continuous-time modulated signal. The demodulator incorporates a so-called slicer or decision device which applies a series of decision thresholds to generate an estimate of the transmitted symbols and so reconstruct the original bit stream.
The carrier frequency is generated in the transmitter from a local timing reference such as a crystal oscillator. Coherent demodulation of a passband signal in the receiver requires exactly the same carrier frequency and phase to perform the demodulation, but the receiver usually has an independent timing reference.
In practice the continuous-time signal received at the demodulator will have been corrupted by the channel (which includes the effects of modulation, the transmission medium, and the receiver). One effect of this corruption will manifest itself as a phase and/or frequency offset, that is to say the phase and/or frequency of the demodulated signal may be shifted relative to the original data signal at the transmitter end.
In order to reduce offset it is possible to use a very stable local oscillator such as an oven crystal, but these are very costly and relatively bulky and furthermore do not compensate for offset due to other elements in the channel.
One technique for determining overall phase offset is to employ a transmitter simulator in the receiver and to use the detected symbols emanating from the demodulator to create an estimate of the ideal phase trajectory (only available at the real transmitter) and to compare this with the measured phase trajectory in the receiver. The difference between the two trajectories represents the phase error. The disadvantage of this technique is that it requires substantial amounts of computation and associated hardware.
In the book entitled "Digital Communication" by Lee and Messerschmitt, published by Kluwer Academic Publishers at pages 549-554, there is proposed a technique which can track phase and frequency offset, in which individual detected symbols emanating from the demodulator are used to determine the phase error.